Monotone Response for Random Objects

2026-08-01

AI Summary

This study addresses the limitation of existing monotonicity assumptions—such as monotone treatment response (MTR), monotone treatment selection (MTS), and monotone instrumental variables (MIV)—which are confined to partial identification of counterfactual means with scalar outcomes and do not readily extend to random objects in general metric spaces. The authors propose a unified partial identification framework by isometrically embedding arbitrary metric spaces into an L² space and imposing coordinate-wise monotonicity on the embedding functions. This approach generalizes MTR, MTS, and MIV assumptions to settings involving random objects, enabling sharp identification of Fréchet means, Wasserstein distributions, interval-valued data, and compositional data without requiring support restrictions. Empirical applications to Job Corps earnings data and NHANES periodontal health distributions demonstrate the framework’s validity and practical utility.

Abstract

Monotone treatment response (MTR), monotone treatment selection (MTS), and monotone instrumental variable (MIV) assumptions are widely used to partially identify counterfactual mean outcomes, but existing analyses have focused almost exclusively on scalar outcomes. We develop a unified framework for partial identification with outcomes that take values in a general metric space under these monotonicity restrictions by embedding the metric space into an $L^2$ space and imposing coordinatewise monotonicity on the embedded functions. The proposed framework yields valid identified sets for Fréchet means in a broad class of random-object spaces and further delivers sharp identification results for distributional outcomes under the Wasserstein metric, interval-valued outcomes represented by support functions, and compositional outcomes under the Aitchison metric. We also establish a support-free characterization of the identified set under the joint MTR--MTS assumption. Numerical and empirical illustrations based on Job Corps earnings data and periodontal health distributions from the National Health and Nutrition Examination Survey demonstrate the empirical usefulness of the proposed framework.

Problem

• monotone treatment response
• partial identification
• random objects
• metric space
• Fréchet mean

Innovation

• monotone treatment response
• partial identification
• random objects
• metric space embedding
• Fréchet mean